China tried to build a wall to keep the Mongols out. They build a great wall, but they didn't keep the Mongols out. They manned it militarily, still didn't keep the Mongols out.
I mean, what would it cost for us to do what you suggest? It would cost a lot of money to monitor the entire border, it would cost a lot of money to hire the agents, and it would cost our economy a lot to not have the laborers (good chances are those fruit pickers would be replaced by illegal child workers, or it would cause the cost of fruit to go up, something our obese country does not need). I think it would be inhumane of us to not give health care to people who need it; when they're actually hurt, and in the ER, not to help them is wrong (and I believe against the Hippocratic oath). Not giving free education could deter some incentive, but it could also just make those who do come here all that less associated with the culture they're in, and all the less educated, making the problem even worse for us. Plus, this issue doesn't deal with "anchor babies," otherwise we're denying education to American citizens, and blaming kids for their parents mistakes.
The reason people come here isn't because they just want to live in America, but becuase their home conditions are so shitty, that they see it as a matter of life and death to get here anyways. Shooting them for crossing the boarder may prevent some people from coming, but for many, they're going to take the risk because they have to feed their families, or take care of their sick and dying mother. How long before they fight back? There's a ton of guns in Mexico, so maybe all we'd do is make the situation more violent, and more warlike.
Also, many illegal immigrants come to this country legally, then overstay their welcome. Stricter boarder control wouldn't address that issue, becuase they're not crossing the boarder illegally.